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How can I prove that the Grassmannian manifold $Gr(k,n)$ is a holomorphic manifold? For example I'd like to prove it for $Gr(2,4)$. How can I do it finding charts and proving that changes of chartes are holomorphic?

Arctic Char
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ArthurStuart
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    One way to do this is to use affine coordinates. See here: http://books.google.ca/books?id=_XxZdhbtf1sC&pg=PA66&lpg=PA66&dq=affine+coordinates+grassmannian&source=bl&ots=3m6lHfpIkz&sig=_FNuzAw73D8r1MaedBCTmqE7Ycc&hl=fr#v=onepage&q=affine%20coordinates%20grassmannian&f=false – Julien Feb 04 '13 at 17:56
  • @julien Is there a more elegant way? Using Segre map...? – ArthurStuart Feb 04 '13 at 18:01
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    How did you prove that it id a manifold? (Or an affine variety)? If the transition functions you had are not regular functions I would be very surprised! (I gaave an atlas here by the way) – Mariano Suárez-Álvarez Feb 04 '13 at 18:07
  • I have to prove that it is a manifold... so I have to find the charts and show that changes of charts are olomorphic. – ArthurStuart Feb 04 '13 at 18:12
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    @ArthurStuart Of course, it is a matter of personal taste, but I find the affine coordinates fairly elegant. – Julien Feb 04 '13 at 18:17
  • But when I find affine coordinates I have to prove that the changes of coordinates are holomorphic... – ArthurStuart Feb 04 '13 at 18:23
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    Have you worked out the changes of coordinates? I'd expect them to be rational functions, so they'll be holomorphic wherever the denominators don't vanish. (And where the denominators do vanish will be outside the relevant charts.) – Andreas Blass Mar 05 '13 at 20:40

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